Hidden Triples and Quads

There is a hidden triple when, in a unit (row, column or box), three numbers can only go in the same three cells. Each of the three goes in one of them, and there is no room left for anything else: the other candidates in those three cells are removed.

It is the hidden pair with one more number and one more cell, and the naked triple seen from the other side. As with the naked triple, none of the three numbers needs to appear in all three cells: it is enough that, together, they don't go outside them.

An example

1. There are no singles, locked candidates, pairs or naked triples in this grid. Row 5 is missing seven numbers: the 1, 2, 3, 4, 6, 8 and 9. These are the notes of the seven empty cells.
2. Let's look for the 3, the 6 and the 8. The 3 only appears in columns 5 and 6; the 6, in columns 2 and 6; the 8, in columns 2 and 5. The three numbers only fit in three cells: those in columns 2, 5 and 6.
3. The 3, 6 and 8 of row 5 have to go in these three cells, one in each, and there is no room left for another number. The 2 and the 9 go from the cells in columns 2 and 5, and the 1 and the 4 from the cell in column 6.
4. In column 6, the 1 could go in rows 5 and 9. Without the one in row 5, it is left with a single place, in row 9: a hidden single.

After the clean-up, the three cells are left with only the 3, 6 and 8: the hidden triple has become a naked triple. The two techniques describe the same fact. The hidden triple lets you recognise it sooner, while the cells still have other notes.

Naked or hidden

In a unit, subsets come in pairs: if there is a hidden one, the other empty cells form a naked one, and vice versa. Both allow the same removals. In the example, the other four cells of row 5 have, between them, only the 1, 2, 4 and 9: they are a naked quad, which removes the 1, 2, 4 and 9 from the three cells of the triple. It is worth looking for the smaller of the two, because it is easier to see.

Hidden quads

Four numbers that only fit in the same four cells form a hidden quad. It is the rarest Hard technique. Almost always, the other empty cells of the unit form a smaller naked subset, which gets to the same result and is seen first. The exception is a unit with no numbers yet: there the other side would have five cells, and the hidden quad is the shortest way.

How to search

• Number by number, one unit at a time. For each missing number, note the cells where it can go, as with the hidden pair.
• Look for three numbers that, together, only take up three cells. Each may fit in two or three cells; what counts is that, together, they don't go beyond three.
• Check whether it removes anything. If the three cells only have those three numbers, the triple is already naked and there is nothing to remove.
• After removing, look for singles, especially in the rows, columns and boxes that the three cells cross.

Your turn

Find a hidden triple and cross out the other candidates in its three cells, leaving only the three numbers. Tap each cell, choose the number and press «Check».